To meter the energy, an energy metering system usually needs to sense the current that is being dissipated by the load. Another way to meter the energy is to sense di/dt (the time derivative of the current) by a Rogowski coil. The flux density of a magnetic field induced by a current is directly proportional to the magnitude of the current. The changes in the magnetic flux density passing through a conductor loop generate an electromotive force (EMF) between the two ends of the loop. The EMF is a voltage signal proportional to the di/dt of the current. The voltage output from the di/dt current sensor is determined by the mutual inductance between the current carrying conductor and the di/dt sensor [1]. Assuming the mutual inductance is M, the EMF is given by
      ⅇ    ⁡          (      t      )        =            -      M        ⁢                  ⅆ                  i          ⁡                      (            t            )                                      ⅆ        t            
In the frequency domain, ignoring the scaling factor, the transfer function of the Rogowski coil is given byHr(s)=s 
The current signal needs to be recovered from the di/dt signal before it can be used. An integrator is therefore necessary to restore the current signal to its original form. Its transfer function is given by
            H      i        ⁡          (      s      )        =      1    s  
Therefore, when the current goes through the Rogowski coil and the ideal integrator, its overall transfer function becomes
                    H        r            ⁡              (        s        )              ⁢                  H        i            ⁡              (        s        )              =            s      ·              1        s              =    1  and the signal is restored to its original form without any distortion. The realization of the ideal integration function
            H      i        ⁡          (      s      )        =      1    s  directly determines the accuracy of the metering system.
Since there is a pole at the origin in the transfer function
                    H        i            ⁡              (        s        )              =          1      s        ,the ideal analog integration function is not stable. The actual implementation of
            H      i        ⁡          (      s      )        =      1    s  is an approximation by moving the pole to the left-hand side in the S-domain. The error of this approximation is referred to as the approximation error. An example is to modify
                    H        i            ⁡              (        s        )              ⁢                              ⁢                            ⁢    to    ⁢                  ⁢                  H        i        ′            ⁡              (        s        )              =                              1                      s            -            a                          ⁢        where        ⁢                                  ⁢        a            ≥              0.        ⁢                                  ⁢                              H            i            ′                    ⁡                      (            s            )                                =          1              s        -        a            is generally known as a lossy integration function. The larger the lossy factor α is, the more stable the integrator becomes, but at the same time the amplitude and phase errors increase compared to those of the ideal integration function
            H      i        ⁡          (      s      )        =            1      s        .  It is difficult to minimize the approximation error for a wide frequency band of interest.
Another way to realize the ideal analog integration function
            H      i        ⁡          (      s      )        =      1    s  is to convert the signal into the digital format after the Rogowski coil and use digital circuits to realize the integration function. The method has two sources of errors compared to the analog implementation of the ideal integration function having only the approximation error.
First, the mapping from the S-domain to the Z-domain (from the analog to the digital) introduces error due to digitization of both the amplitude and the time step. This is referred to as the mapping error. Typical first order integration functions have the following three forms.
Forward integration function:
            H      i        ⁡          (              z                  -          1                    )        =            z              -        1                    1      -              z                  -          1                    
Backward integration function:
            H      i        ⁡          (              z                  -          1                    )        =      1          1      -              z                  -          1                    
Bilinear integration function:
            H      i        ⁡          (              z                  -          1                    )        =            1      2        ·                  1        +                  z                      -            1                                      1        -                  z                      -            1                              
As the signal frequency relative to the digitization frequency increases, the error increases due to the fact that the time step relative to the signal is increased. The second arises from the implementation of the digital integration function Hi(z−1), which usually has at least one pole on the unit circle. As in implementing the ideal analog integration function, the actual implementation of the Hi(z−1) involves moving the pole inside the unit circle. This results in an approximation error in the same fashion as that in implementing the ideal analog integration function. This approximation error usually increases as the signal frequency increases. This is evident in [1]. Both the amplitude and the phase response are only the approximation to the digital integration function.
Even though there are two error sources (the mapping error and the approximation error) for the digital implementations, the digital implementation is generally preferred due to the fact that the analog implementation is difficult to be integrated into a microchip. The major hurdle for integration is the large on-chip resistance and capacitance as well as their large variations as a function of processing and temperature.
Both the mapping and approximation errors increase as the signal frequency increases. In a metering system, more and more harmonics are required to be metered, therefore the accuracy in metering higher order harmonics is of significance.
The invention is conceived to eliminate the approximation error when realizing the ideal analog integration function and the digital integration function. The method also provides means to minimize the mapping error in the digital realization by utilizing higher order digital integration function.